Date/Time : 24 Jan 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Wang Kelei, Wuhan University

Since the seminal work of Alt-Caffarelli in 1981, the one phase free boundary problem has been studied by many people. To study the regularity and singularity of free boundaries, blow up analysis is a standard method. It turns out for this free boundary problem, the nondegeneracy condition is crucial for the application of this method. Although this condition holds for energy minimizers, it’s not true for general solutions. In this talk, I will discuss a proof of the nondegeneracy for stable solutions. This is based on a joint work with N. Kamburov.

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Date/Time : 18 Jan 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Ruixiang Zhang, University of California, Berkeley

The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\”{o}rmander asked if a more general class of operators (known as H\”{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\”{o}rmander’s question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\”{o}rmander’s question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a “positive curvature” assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it “Bourgain’s condition”. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\”{o}rmander type operator satisfying Bourgain’s condition should have the same $L^p$-boundedness as in those two conjectures. As evidences, we prove that the failure of Bourgain’s condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

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Date/Time : 29 Nov 2023 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jaemin Park, University of Basel

In this talk, we study stationary solutions to the 2D incompressible Euler equations in the whole plane. It is well-known that any radial vorticity is stationary. For compactly supported vorticity, it is more difficult to see whether a stationary solution has to be radial. In the case where the vorticity is non-negative, it has been shown that any stationary solution has be radial. By allowing the vorticity to change the sign, we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or finite. For the finite energy case, it turns out that a construction of a stationary solution with compactly supported velocity is possible. This is based on the joint work with Javier Gomez-Serrano and Jia Shi.

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Date/Time : 29 Nov 2023 15:00
Venue : S17 #06-11 (SR6)
Speaker : Dawei Shen, Sorbonne University

The global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou-Klainerman in 1993. In 2007, Bieri has extended the result of Christodoulou-Klainerman under lower decay and regularity assumptions on the initial data. In this talk, we will introduce a recent work, which extends the result of Bieri to minimal decay assumptions.

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Date/Time : 08 Nov 2023 14:00
Venue : S17 #05-11 (SR5)
Speaker : Xiaoqian Xu, Duke Kunshan University

In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of application is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and the role it plays in the field of advection-diffusion-reaction equation, such as the famous Keller-Segel equation for chemotaxis. I will also discuss about the examples of such flows in this talk.

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Date/Time : 19 Oct 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

The studies of Poincaré-Einstein manifolds originate from the AdS/CFT correspondence in mathematical physics, which relates quantum gravity theory and conformal field theory. On the mathematics side, numerous conformal invariants were successfully constructed via the microlocal analysis on Poincaré-Einstein spaces, so that it has become a central area in conformal geometry.

A primary object in this talk is the so-called fractional GJMS operators, which provide a uniform treatment of conformally covariant operators with different orders. From the analytic point of view, the degenerate and nonlocal equations arising in this setting are particularly interesting. We will mainly exhibit a series of sharp dimension bounds and Hausdorff measure estimates for the singular set of those solutions. The proof is based on the quantitative differentiation theory for degenerate elliptic operators and involves subtle regularity issues on the Poincaré-Einstein metrics.

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Date/Time : 11 Oct 2023 14:30
Venue : S17 #05-11 (SR5)
Speaker : Chen Shibing, University of Science and Technology of China

Optimal partial transportation is an important generalization of the classical optimal transport problem, and it arises in many applications, such as image retrieval. Mathematically, the problem was first studied by Caffarelli and McCann. They established the existence, uniqueness, and Holder regularity of the free normal and the map near the free boundary while leaving open the question of their higher regularity. In this talk, we will introduce our recent work in this direction.

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Date/Time : 04 Oct 2023 14:30
Venue : S17 #05-11 (SR5)
Speaker : Shiwu Yang, Peking University

It has been shown that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space decay like linear solutions. One hence can define the associated radiation field on the future null infinity. In this talk, we show the existence of a global solution to the MKG system which scatters to any given sufficiently localized radiation field with arbitrarily large size and total charge. The result follows by studying the characteristic initial value problem to the MKG system with general large data by using gauge invariant vector field method. This is based on joint work with W. Dai, H. Mei and D. Wei.

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Date/Time : 17 Aug 2023 13:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Fei, Shanghai Jiaotong University.

This paper concerns the existence of global weak solutions {\it \`a la Leray} for compressible Navier–Stokes–Fourier system with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.

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